#72250318

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#18194
commutator, Frattini, cosocle	C₂²×M₁₆	C₂⁴
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(8,20)(12,26)(17,34)(27,33);(2,4)(6,11)(7,16)(8,12)(10,21)(13,28)(14,31)(15,24)(18,19)(20,26)(22,32)(23,25)(27,33)(29,35)(30,36);(5,6,12,25,10,14,20,24,9,15,26,31,21,23,8,11)(7,13,27,36,19,29,34,35,18,30,33,28,16,32,17,22);(5,7,10,19,9,18,21,16)(6,13,14,29,15,30,23,32)(8,17,12,27,20,34,26,33)(11,22,25,36,24,35,31,28)

Polycyclic presentation:

2^10|c^2=ei,d^2=i,e^2=ij,i^2=j,b^a=bg,c^a=cf,c^b=ce,d^b=dij,e^a=eh,e^b=ei,f^b=fhj,f^c=fh,f^e=fj,h^c=hj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250326, #72250381, #72257258, …
maximal normal subgroup of	#72250326, #72250381, #72257258, …
maximal quotient of	#72308136

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.5*F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);